Poroviscoelasticity and compression-softening of agarose hydrogels

Abstract

Agarose hydrogels are poroviscoelastic materials that exhibit a waterlogged-crosslinked microstructure. Despite an extensive use in biotechnologies and numerous studies of the elastic properties of agarose gels, little is known about the compressible behavior and the microstructural changes of such fibrillar hydrogels under compression. The present work investigates the mechanical response of centimeter-sized pre-molded agarose cylinders when applying a compressive strain ramp over an extended range of loading speed and polymer concentration. One of the original contributions is the simultaneous monitoring of the changes in the hydrogel volume to determine the Poisson’s ratio through a spatiotemporal method. The linear poroelastic response of agarose hydrogels shows a compressible behavior at strain rates less than 0.7 % s⁻¹. The critical compressive strain of a few percent at the onset of the non-linear regime and the always positive Poisson’s ratio decrease when applying a slow compressive ramp. The mechanical response in the linear regime is typical of a deformation mode either dominated by the bending of semiflexible strands (enthalpic regime) or by the stretching of the network (entropic regime) at higher agarose concentration. Cyclic linear shear deformations superimposed to a compressive strain from 0.5 up to 40% further give evidence of a compression-softening of the network causing the transition to the non-linear regime without dependence upon the network topology and connectivity. Finally, the buckling-induced aging of the network under a weak compression and the poroviscoelasticity of the hydrogel are shown to impact the relaxation of the normal stress and the equilibrium stress.

Appendix

Mean pore diameter of agarose hydrogels

Cryo-SEM micrographs of agarose hydrogels show random fibrillar networks with a typical mean pore diameter d decreasing with the mass concentration of the polymer from a micrometer size at c ≈ 0.3 wt% down to a few hundreds of nanometers at c ≈ 3 wt% (Fig. 2). The pore size distribution is broad with a factor of about 10 between the mean smaller diameter d_min and the mean larger diameter d_max of pores in the network (full black and red circles in Fig. 16). Mechanical-scanning probe microscopy indeed gives evidence of the inhomogeneous microstructure of agarose hydrogels at low concentrations 0.5 wt %  < c < 2 wt% with a log-normal distribution of the local shear modulus, while a simple Gaussian distribution describes the histogram of local elastic moduli values in thermal gels made of flexible chains such as polyacrylamide (Nitta et al. 2003). As expected for a fibrillar semiflexible network with a mesh size larger than the strand diameter, the mean pore diameter of agarose hydrogels scales as (c − c_g)^λ with λ ≈  − 0.5 at 0.3 wt%  < c < 3 wt% whatever one considers either the larger or the smaller free spaces (Fig. 16). Note that the scaling exponent λ ≈ − 0.5 takes slightly more negative values when considering the mass concentration in place of the deviation concentration c − c_g from the percolation threshold (Maaloum et al. 1998; Righetti et al. 1981). Atomic force microscopic observations from Maaloum et al. (1998) or studies of the electrophoretic mobility of latex particles in agarose hydrogels from Righetti et al. (1981) indeed lead to a similar scaling exponent λ ≈ − 0.5 provided that the mean pore diameter is plotted versus the deviation concentration c − c_g from the percolation point (Fig. 16). The length scale explored in sparsely connected networks is further very sensitive to the choice of the investigation method due to the broad pore size distribution of biopolymer hydrogels.

Fig. 16

Logarithmic plot of the mean pore diameter d versus the agarose mass concentration c - c_g with a gelation threshold c_g = 0.1 wt%. Full black circles and red circles refer to the mean smaller diameter d_min and the mean larger diameter d_max of pores extracted from Cryo-SEM micrographs of Setexam hydrogels at polymer mass concentrations c ≈ 0.3 wt% (L0.3), c ≈ 0.99 wt% (L1) and c ≈ 2.9 wt% (L3). Full green squares and full blue squares, respectively, stand for the experimental data of Maaloum et al. (1998) and Righetti et al. (1981). Full color lines are the best power law fits of (d, c − c_g) data and the dotted black line highlights the common slope of the power law fits with an indication of the value of the scaling exponent λ ≈ − 0.5

Compression–tension of agarose hydrogels

The reversibility of the hydrogel response is studied by applying a compression–tension cycle at a speed dh/dt = 100 µm/s with a maximum compressive strain ε_m of either 2% (black curves in Fig. 17) or 10% (red curves in Fig. 17). The stress–strain response appears as nearly reversible in the linear regime for a maximum compressive ε_m ≈ 2% less than the critical strain ε_c as water exudation remains negligible during a fast compression–tension cycle (black curves in Fig. 17). A more detailed examination of σ(ε) curves nevertheless shows a weak hysteresis and a slope (dσ/dε)(ε = ε_m) less important at the end of the compressive ramp compared to the beginning of the tension ramp, especially at low agarose concentration (L1 sample in Fig. 17a). The asymmetry observed in the response of agarose hydrogels to a compression–tension cycle in the linear regime likely results from the deformation mode of strands since it is easier to bend a fiber during compression than to stretch the fiber during extension.

Fig. 17

Normal stress σ versus the compressive strain ε of L1 (a), L3 (b), and L9 (c) hydrogels when applying a compression–tension cycle at a loading speed dh/dt = 100 µm/s with a maximum amplitude ε_m ≈ 2% (black curves) or ε_m ≈ 10% of the strain ramp (red curves). Dashed vertical full lines show the critical compressive strain ε_c delimiting the transition to the non-linear regime. Setexam agarose hydrogels with polymer mass concentration c ≈ 0.99 wt% (L1), c ≈ 2.9 wt% (L3), and c ≈ 8.3 wt% (L9)

As expected, the hysteresis becomes more significant in the non-linear regime for a maximum compressive strain ε_m ≈10 % > ε_c (full red lines in Fig. 17) as the unbuckling of strands and the breaking of some weak bonds require longer timescales during the extension phase. The degree of hysteresis increases somewhat with the polymer concentration (Fig. 17), and the gel cylinder almost recovers the initial height h_o ≈14 mm after a time period from a few minutes to a few hours when removing the load which confirms the dominant elastic behavior of agarose hydrogels. The compression-softening of the fibrillar network thus appears as nearly reversible even for very large compressive strain at very low loading speed as long as the hydrogel remains intact without any stress-induced microfractures. A second compression–tension cycle is necessary to observe a fully reversible deformation of the agarose hydrogel (data not shown) likely as a result of the formation of some extra irreversible bonds between buckled strands during the first compression ramp (paper in preparation).

Agarose molecular weight and elasticity of agarose hydrogels

Compression experiments were performed using the Brookfield texture analyzer and agarose hydrogels prepared at different concentrations 0.5 wt% ≤ c ≤ 9 wt% with two different powders supplied either by Setexam or Sigma (Section 2.1). The stress–strain curves of Sigma and Setexam agarose hydrogels considered as incompressible when applying a fast 15% strain ramp at a loading speed dh/dt = 100 µm/s exhibit similar features (Figs. 10a and 18a). As expected, the agarose average molecular weight weakly influences the scaling exponents β ≈ 2.15 ± 0.05 or β ≈ 1.36 ‐ 1.41 representative of either the enthalpic or the entropic elasticity (Fig. 18b). However, Fig. 18b gives evidence of higher values of the Young modulus E_o of Sigma agarose hydrogels (M_w ≈ 3.05 10⁵ g/mol) compared to that of Setexam agarose hydrogels (M_w ≈ 1.88 10⁵ g/mol) regardless of the agarose concentration. The power law fits of the fiber volume fraction dependence of the Young modulus E_o(ϕ - ϕ_g) of Sigma and agarose hydrogels in the enthalpic and entropic elastic regimes, respectively, give ${E}_o\approx {M}_w^{0.8}$. The Young modulus of biopolymer fibrillar networks is usually reported to scale as ${E}_o\approx {M}_w^2$ for lower average molecular weight 3 10⁴ g/mol < M_w < 7 10⁴ g/mol (Eldridge and Ferry 1954). The dependence on M_w  of the elastic modulus of agarose hydrogels is expected to be lower for higher average molecular weight and higher polymer concentrations as the number of dangling ends not involved in the elasticity of the network becomes less (Normand et al. 2000).

Fig. 18

(a) Normal stress σ versus the compressive strain ε for Hα hydrogels in water when applying a 15% strain ramp at a loading speed dh/dt = 100 µm/s in the Brookfield texture analyzer. The insert is a zoom in the low-strain region where dotted lines are best linear fits of σ(ε) data in the limit of low compressive strain ε < ε_c and dashed vertical lines show the critical compressive strain ε_c. (b) Semilogarithmic plot of the Young modulus E_o of the hydrogel assumed as incompressible versus the fiber volume fraction ϕ - ϕ_g with ϕ_g = 0.1% either for Sigma (square symbols) or Setexam (circle symbols) agarose cylinders rapidly compressed in air (open symbols) or in water (full blue symbols). Full black and grey lines are the best power law fits of (E_o, ϕ - ϕ_g) data for Sigma and Setexam agarose hydrogels, respectively, in the concentration regimes 0.4 % ≤ ϕ - ϕ_g ≤ 2.8 % and 3.75 % ≤ ϕ - ϕ_g ≤ 8.2%. Dotted black lines and dotted grey lines highlight the respective slopes of power law fits with an indication of the value of the elastic exponent β either in the enthalpic or in the entropic elastic regime

Viscoelasticity of agarose hydrogels

The shear stress relaxation response of quasi-equilibrium agarose hydrogels was monitored in the linear regime when applying a low compressive strain ε ≈ 2% (Section 2.4). The characteristic viscoelastic time t₂ at short timescale decreases from 2 min down to 16 seconds when increasing the agarose concentration 0.5 wt%  < c < 7 wt% as the hydrogel becomes stiffer (Fig. 19). The retarded elastic response (G₂ + G₃)/G₀ of about 10% to 15% for agarose concentration c > 1 wt% surprisingly takes on a much larger value (G₂ + G₃)/G₀ ≈ 48% for a diluted 0.5 wt% agarose hydrogel (L0.5 sample in Fig. 19 and Table 4). Such a sharp increase in the apparent viscoelasticity of a diluted agarose hydrogel probably results from the buckling-induced aging of the sparsely connected network on long timescale and the gradual emergence of non-linear effects.

Fig. 19

Time relaxation of the dimensionless shear stress modulus G(t)/G_o of quasi-equilibrium hydrogels L0.5 (grey color), L1 (green color), L3 (blue color), and L7 (brown color) weakly compressed in water when superimposing a shear strain in the linear regime. Before the stress relaxation test, a fast 4% compressive strain ramp is imposed at a loading speed dh/dt = 100 µm/s and the static strain ε = 4% is hold for 15min. Dotted red curves correspond to a second-order generalized Maxwell model (Eq. (1)) fitted to data with the parameter values shown in Table 4. Setexam agarose hydrogels with a polymer mass concentration c ≈ 0.5 wt% (L0.5), c ≈ 0.99 wt% (L1), c ≈ 2.9 wt% (L3), and c ≈ 6.5 wt% (L7)